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Despite exhibiting the highest melting point of all metals, the technological use of tungsten is hampered by its room-temperature brittleness. Alloying with Re significantly ductilizes the material which has been assigned to modified properties of the screw dislocation. Using density functional theory, we show that alloying induces a transition from a symmetric to an asymmetric core and a reduction in Peierls stress. This combination ductilizes the alloy as the number of available slip planes is increased and the critical stress needed to start plastic deformation is lowered.